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Mirrors > Home > MPE Home > Th. List > sucexeloni | Structured version Visualization version GIF version |
Description: If the successor of an ordinal number exists, it is an ordinal number. This variation of onsuc 7831 does not require ax-un 7754. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof shortened by BJ, 11-Jan-2025.) |
Ref | Expression |
---|---|
sucexeloni | ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6396 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ordsuci 7828 | . . 3 ⊢ (Ord 𝐴 → Ord suc 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ On → Ord suc 𝐴) |
4 | elex 3499 | . 2 ⊢ (suc 𝐴 ∈ 𝑉 → suc 𝐴 ∈ V) | |
5 | elong 6394 | . . 3 ⊢ (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴)) | |
6 | 5 | biimparc 479 | . 2 ⊢ ((Ord suc 𝐴 ∧ suc 𝐴 ∈ V) → suc 𝐴 ∈ On) |
7 | 3, 4, 6 | syl2an 596 | 1 ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 Vcvv 3478 Ord word 6385 Oncon0 6386 suc csuc 6388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 df-suc 6392 |
This theorem is referenced by: onsuc 7831 1on 8517 2on 8519 |
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